Let f(x) be a nonconstant polynomial in x with integer coefficients and suppose that for five distinct integers a1, a2, a3, a4, a5, one has f(a1)= f(a2)= f(a3)= f(a4)= f(a5)= 2.
Find all integers b such that f(b)= 9.
If f(x)=2 at a1, a2, a3, a4, and a5, then g(x)=f(x)2 has those roots, and can be written as (xa1)(xa2)(xa3)(xa4)(xa5)h(x), where h(x) is another polynomial with integer coefficients.
If f(x)=9, then g(x)=7... but for integer x, g(x) is the product of at least 5 different integer values, and we cannot write 7 that way, for any values of a1 thorugh a5.
For the extra question by Federico Kereki, we are looking for a square S such that S2 can be written as the product of at lest 5 different integer values. I found 100 might do the job, for 98=(7).(2).(1).(1).(7) with h(x)=1, but of course, this would work only for very specific values of a1, a2, a3, a4, and a5: a1=any, a2=a15, a3=a21, a4=a32, a5=a46, and then b=a17.

Posted by e.g.
on 20060526 12:43:32 